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Let $G$ be a finite 2-group of nilpotency class two such that $\frac{G}{Z(G)}=\{Z(G), aZ(G), bZ(G), abZ(G)\}\simeq C_{2}\times C_{2}$. Then do there exist a non inner automorphism $\alpha$ of $G$ such that $\alpha(a)\neq a$, $\alpha(b)\neq b$ and $\alpha(ab)\neq ab$ ? For example this true for $D_{8}$, dihedral group of order 8, or $Q_{8}$, generalized quaternion group of order 8.

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Could you say something about how you came across this question, because as written it looks like a homework question. –  Noah Snyder Jun 7 '12 at 5:12
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Derek Holt answered this question 2 hours ago at math.stackexchange: math.stackexchange.com/a/155023/669 –  j.p. Jun 7 '12 at 11:19
    
Noah, it seems like it might not be homework, just yet another case (if you look at the OP's original question) of someone wanting information rather than thinking for themselves –  Yemon Choi Jun 7 '12 at 17:15
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